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In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function. The original examples of distributions occur, unnamed, as functions φ on Q/Z satisfying〔Kubert & Lang (1981) p.1〕 : We shall call these ordinary distributions.〔Lang (1990) p.53〕 They also occur in ''p''-adic integration theory in Iwasawa theory.〔Mazur & Swinnerton-Dyer (1972) p. 36〕 Let ... → ''X''''n''+1 → ''X''''n'' → ... be a projective system of finite sets with surjections, indexed by the natural numbers, and let ''X'' be their projective limit. We give each ''X''''n'' the discrete topology, so that ''X'' is compact. Let φ = (φ''n'') be a family of functions on ''X''''n'' taking values in an abelian group ''V'' and compatible with the projective system: : for some ''weight function'' ''w''. The family φ is then a ''distribution'' on the projective system ''X''. A function ''f'' on ''X'' is "locally constant", or a "step function" if it factors through some ''X''''n''. We can define an integral of a step function against φ as : The definition extends to more general projective systems, such as those indexed by the positive integers ordered by divisibility. As an important special case consider the projective system Z/''n''Z indexed by positive integers ordered by divisibility. We identify this with the system (1/''n'')Z/Z with limit Q/Z. For ''x'' in ''R'' we let ⟨''x''⟩ denote the fractional part of ''x'' normalised to 0 ≤ ⟨''x''⟩ < 1, and let denote the fractional part normalised to 0 < ≤ 1. ==Examples== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Distribution (number theory)」の詳細全文を読む スポンサード リンク
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